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Quantum from principles
1) Pure states
2) Transformations
3) Measurements
4) Probabilities
5) Composition
Quantum theory
3
Introduction
● Quantum theory can be defined via a short number of clear postulates.
● Although these postulates are strictly speaking independent, one can ask questions of the following type:
● “Are there theories with the same pure states as quantum theory?”
4
Introduction
● Need a general framework to talk about theories.● What type of theory is quantum theory?● Is it an information theory? A logic? A form of
mechanics? A general probability theory? A process theory?
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Introduction
● Are there logics with the same measurement structure as quantum theory but different states?
● Are there operational theories with the same pure states and dynamics as quantum theory but different measurements?
● The answers to these questions will tell us how independent the postulate are.
6
Introduction
● Is it possible to change one part of quantum theory whilst keeping the others constant?
● Could we obtain a future theory of physics by making “perturbations” to the current theory? Or will we need a radical overhaul?
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Introduction
● If we assume quantum theory is a form of logic, then the structure of measurements fully encodes the probability rule.
● Assuming it is a GPT means that the dynamical part fully determines the probabilistic part.
● Adopting different perspectives on quantum theory will allow for different insights into what it has to tell us.
● I don’t argue that any of the perspectives I adopt today are the final word, rather they are useful stepping stones.
Quantum logic and Gleason’s theorem
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Mathematical Foundations of Quantum Mechanics, Von Neumann (1932)
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Quantum logic
● Facts about quantum systems do not obey the rules of classical logic
● Need a new set of rules (a quantum logic) to reason about propositions
Classical logic
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Classical logic (propositional)
● A set of rules about propositions● Propositions can be true or false● Can connect propositions together to form
formulae using connectives: “and” , “or”● One connective is implication
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Classical logic (propositional)
● A set of rules about propositions● Propositions can be true or false
P Q P or Q
T T T
T F T
F T T
F F F
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Classical logic
● A set of rules about propositions● Propositions can be true or false
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Classical logic as set theory
p
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Classical logic as set theory
p
p = being red = all red objects
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Logical connectives in set theoretic language
p q
p = being red = all red objects
q = being square = all square objects
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Logical connectives in set theoretic language
● “and” is set intersection
p q
p = being red = all red objects
q = being square = all square objects
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Logical connectives in set theoretic language
● “and” is set intersection
p q
p and qp = being red = all red objects
q = being square = all square objects
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Logical connectives in set theoretic language
● “or” is set union
p q
p or qp = being red = all red objects
q = being square = all square objects
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Logical connectives in set theoretic language
● “or” is set union
p q
p or q
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Classical logic as set theory
p = “particle is in the blue circle”
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Logical connectives in set theoretic language
● “not” is complement set
not p = “particle is not in the blue circle”
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Logical connectives in set theoretic language
● What is implication?● All things p are also q● “Being a square implies having four sides”● All squares are four sided
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Implication
q
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Implication
qp
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Implication as set inclusion
● Implication is given by set inclusion
qp
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Classical logic
● Classical physics:
a cb d
p = “particle is in interval [a,c]”
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Classical logic
● Classical physics:
a b c d
q = “particle is in interval [b,d]”
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Classical logic
● Classical physics:
a b c d
p and q = “particle is in interval [b,c]”
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A simple example
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A simple example
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A simple example
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Logic as set theory
● Propositions correspond to sets.● Implication amongst propositions given by set
inclusion.● Set inclusion is a partial order:
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Implication
qp
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Implication
qp
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Lattice
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Complete lattice
● Complete lattice: Partially ordered set where every pair of elements has a “and” and “or” defined (meet and join)
● Meet: greatest lower bound● Join: least upper bound
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Complete lattices
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Complete lattices
Meet? Greatest lower bound
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Complete lattices
Meet? Greatest lower bound
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Complete lattices
Meet? Greatest lower bound
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Complete lattices
Join? Lowest upper bound
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● A logic is just a complete lattice● Classical logic is distributive lattice● p and (q or r) = (p and q) or (p and r)● p or (q and r) = (p or q) and (p or r)
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● A logic is just a complete lattice● Classical logic is distributive lattice● Quantum logic is a non-distributive lattice● We have a vantage point to talk about general
non-classical logics, of which quantum logic is an example.
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Quantum logic
● Propositions correspond to subspaces of H.● Examples: , ,● Quantum logic is a lattice of subspaces
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Quantum logic
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Quantum logic
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Quantum logic
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Quantum logic
● Propositions correspond to subspaces of H.● Implication given by inclusion of subspaces
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Quantum logic
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Quantum logic
● Propositions correspond to subspaces of H.● “and” is given by intersection
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Quantum logic
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Quantum logic
● Propositions correspond to subspaces of H.● “or” is given by span
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Quantum logic
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Quantum logic
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Quantum logic
● Propositions correspond to subspaces of H.● Negation is given by orthogonal subspace
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Quantum logic
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Quantum logic
● Propositions correspond to subspaces of H.● Implication given by inclusion of subspaces● “and” is given by intersection ● “or” is given by span● Negation is given by orthogonal complement
(orthocomplemented lattice)
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Quantum logic
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Non-distributivity of quantum logic
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Non-distributivity of quantum logic
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Non-distributivity of quantum logic
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Non-distributivity of quantum logic
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Non-distributivity of quantum logic
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Non-distributivity of quantum logic
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Non-distributivity of quantum logic
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Non-distributivity of quantum logic
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Operational quantum logic
● Propositions (subspaces) can be thought of as answers to yes/no questions (Mackey).
● These questions are the (projective) measurements we carry out.
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Gleason’s theorem
● If the answers correspond to subspaces, what are the possible probability assignments?
● Measure
● For mutually exclusive answers we expect probabilities to add:
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Gleason’s theorem
● If the answers correspond to subspaces, what are the possible probability assignments?
● For projectors onto orthogonal subspaces (mutually exclusive properties)
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Gleason’s theorem
The only measures on L(H) (for ) which obey the lattice structure are of the form
Where is a density operator
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Axiomatisations of quantum theory
● Assume the structure of questions and answers● The only probability rule compatible with the
lattice structure of subspaces is the Born rule● Recover the probability rule and states.
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A sketch for a reconstruction
● Assume structure of questions● Use Gleason’s theorem to get states and
probability rule● Use theorem by Wigner to get the unitary group
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Conclusion
● We adopted a certain perspective on quantum theory.
● Namely it is an Operational Quantum Logic● Using this perspective we see that the axioms
about the probability rule (and states) directly follow from the structure of projective measurements.
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Conclusion
● In classical physics experimental propositions can be identified with sets
● In quantum physics experimental propositions can be identified with subspaces
● Both have a lattice structure
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Where next?
● We will adopt a different operational perspective● That of General Probabilistic Theories● Using this perspective we see that the axioms
about the probability rule (and measurements) directly follow from the structure of pure states.
General probabilistic theories
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Operationalism (1927)
https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/significance_2/index.html
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Operationalism
● “We mean by any concept nothing more than a set of operations; the concept is synonymous with the corresponding set of operations.” Bridgman (1927)
● For instance concept of length should reduce to a set of operations one can carry out with a ruler.
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Operationalism
● Special relativity● Theories defined via a small number of
operational principles.● Operational reconstructions of quantum theory● See Lluis Masanes lectures from last summer
school
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General probabilistic theories
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The operational approach to physical theories
● In the operational approach a theory just allows us to make predictions about the outcomes of measurements.
● No claims are made about ontology or underlying physical reality.
● We have access to some physical devices which we can wire together, and the theory allows us to make predictions about what will happen.
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The operational approach to physical theories
● Operational principles will lead to certain mathematical structure
● Mixing: convex structure● Composition of devices: categorical structure
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Primitives
● The primitives are devices: preparation, transformation and measurement devices.
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The operational approach
● The primitives are devices: preparation, transformation and measurement devices.
Preparation choice
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Measurement device
● The primitives are devices: preparation, transformation and measurement devices.
Measurement readoutMeasurement choice
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Prepare and measure
● The primitives are devices: preparation, transformation and measurement devices.
Preparation choice Measurement readoutMeasurement choice
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An example: qubit
Prepare Outcome Measure in Z basis
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Transformation devices
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The operational approach in quantum foundations
● For various different settings of the preparation device we collect the statistics for various measurements.
● We use these statistics to generate the state space and effect space.
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An example: qubit
Prepare Outcome Measure in Z basis
Outcome
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The operational approach in quantum foundations
● The classical world: laboratories, agents and classical probability theory is assumed.
● Fundamentally not about systems, but about devices, and the correlations between classical inputs and outputs of these devices.
● From the collected statistics we derive the state and effect space.
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Classical probability theory (finite dimensional)
wikipedia Getty images
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Classical systems (finite dimensions)
● Consider a ball which can be placed in one of two boxes:
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Classical systems (finite dimensions)
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Classical systems (finite dimensions)
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Preparations
● A preparation consists of putting the ball in one of the boxes and closing them.
● I can choose to put the ball in box 1 with some probability p, and box 2 with probability 1-p
p
1-p
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Preparations
● A preparation consists of putting the ball in one of the boxes and closing them.
● I can choose to put the ball in box 1 with some probability p, and box 2 with probability 1-p
● This is called preparing an ensemble, and is a central assumption of the GPT framework
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Preparing an ensemble
p
1-p
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Measurements
● I give the closed box to someone else● They can carry out a measurement: open the box
to see which partition the ball is in● For a given preparation we repeat this many
times● The measurement has two outcomes: “The ball is
in box 0” and “The ball is in box 1”.
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An example
Prepare
Outcome Measurement: open both boxes
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An example
Prepare Outcome
Measurement: open both boxes p
1-p
Outcome
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States
● For a given preparation outcome 0 will occur with probability p, and outcome 1 with probability 1-p
State just allows us to make predictions.
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States
p
1-p
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States
● A state is a probability distribution over outcomes
● Just like in operational quantum logic
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State space
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State space
Extremal states (pure states)
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State space
Convex hull of extremal points
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Convex combination
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State space
Extremal states (pure states)
Mixed states (describes an ensemble)
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Subnormalised states
● The experimenter can also choose to just send an empty box
●
●
● This will give probability 0 for all measurement outcomes, and hence is represented by
●
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Cone of states
Normalised states
Subnormalised states
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Effects
● How do I obtain the probability p(0|P) of obtaining outcome 0 from the state ?
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Effects
What is probability of outcome 0 or outcome 1 occurring?
What is probability of neither outcome occurring?
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Subnormalised states
● The experimenter can also prepare mixtures of the empty box and preparations with a ball in one of the partition
p
1-p
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Effects
is an effect. It is a linear functional of states (dual vector).
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Effect space
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Effect space
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Effect space
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The trit
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The trit
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Trit state space
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Classical four level systems
● Tetrahedron, embedded in , just drawing the normalised states
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Classical systems
● A classical system with d pure states is a simplex with d extremal points.
● Line segment, triangle, tetrahedron...
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Deriving the quantum state space
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Qubit: deriving the state space
● Pure states● Two outcome measurements
● Can prepare ensembles
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Qubit: deriving the state space
0.25
0.75
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Qubit
Could write a state as a vector of probability distributions over outcomes
●
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Qubit
● Could write a state as a vector of probability distributions over outcomes.
● But would be infinitely long.● And contains redundancies.
●
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Qubit
● Any can be written as a linear combination of 4 functions of the form
● For instance we can have the +X,+Y,+Z and -Z outcome probabilities as the basis functions.
● Could be a SIC-POVM
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The probabilistic representation
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The probabilistic representation
Any outcome probability can be computed as a linear function of the 4 fiducial probabilities.
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Representing ensembles
is the mixed state representing the ensemble P
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Representing ensembles
● Mixed state representing ensemble is obtained by taking convex combination of vectors
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Qubit state space
https://commons.wikimedia.org/w/index.php?curid=5829358
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Probabilistic representation
● This is the representation of states in which ensembles are represented by convex combinations of state vectors.
● For example the Bloch vector representation/density matrix representation.
● This is the representation we use in GPTs.
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Key facts about the probabilistic representation
● Outcome probabilities are linear functions of the states.
● We went from to ● We can take mixtures of states by taking convex
combinations of state vectors.● Since for any pair of states we can take a mixture
the state space is convex.
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Convex sets
● For every pair of points in the set the line segment joining them is part of the set.
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● Classical and quantum state spaces correspond to convex sets.
● Other convex sets correspond to other non-classical systems.
● Like the general lattice structure and non-classical logics.
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General GPT: single system
State space
Effect space
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How to “read” a convex state space
Mixtures of states:●
a b
c
d
0.25
0.75
0.5
0.5
a
b
c
d
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“True” definition of mixed states
● Multiple ensembles are assigned the same mixed state.
● A mixed state is an equivalence class of ensembles.
● See Holevo: “Probabilistic and Statistical Aspects of Quantum Theory”
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Perfectly distinguishable states
a
b
Outcome 0
Outcome 1
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How to “read” a convex state space
● Effects are linear functionals.
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How to “read” a convex state space
● Effects are linear functionals.
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Perfectly distinguishable states
● So antipodal states are perfectly distinguishable.● There is a single measurement where one of the
outcomes occurs with certainty on one of the states, and with probability 0 for the other
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The gbit (one half of a PR box)
● Let us consider a system which can be prepared in 4 pure states, and mixtures of these 4 pure states.
● Its state space is the square.● Like all non-classical state spaces, there are
certain ensembles which are indistinguishable.
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Indistinguishable ensembles
A B
CD
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How to “read” a convex state space
A B
CD
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How to “read” a convex state space
A B
CD
Can perfectly distinguish A,B from C,D
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How to “read” a convex state space
A
Can we perfectly distinguish A from B?
B
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How to “read” a convex state space
A
B
Can we perfectly distinguish A from B?
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How to “read” a convex state space
A
B
Can we perfectly distinguish A from B?
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How to “read” a convex state space
A
B
Can we perfectly distinguish A from B?
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Composite systems
● As well as composing devices in sequence
● Can compose the in parallel
Preparation choice Measurement readoutMeasurement choice
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Parallel composition
A
B
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Operational probabilistic theories
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Composite systems
A
B
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Composite systems (no-signalling)
A
B
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Composite systems (no-signalling)
B
A
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Reduced states
A
B
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Composite systems: state spaces
A ABB
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Product states
A ABB
P:
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Product states: quantum theory
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Product states
● This map is bilinear (due to mixing)● If we assume local tomography it is just a tensor
product● But not in general
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Reduced states
AAB
R:
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Reduced states: quantum theory
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Reduced states
● This map is a linear map from to● Obtained by taking the unit effect on subsytem B
and the identity on A●
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Effect space
● We have equivalent maps on the effect space.
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Conclusion
● Have a framework within which we can talk about operational theories: general probabilistic theories
● Highlighted the convex structure, which arises from the possibility of taking mixtures
● Mentioned the categorical structure which arises from composing devices
● Arbitrary convex sets correspond to non-classical systems
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Redundancy of the measurement postulates of quantum theory
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1) States
2) Transformations
3) Measurements
4) Probabilities
5) Composition
Quantum theory
1) States
2) Transformations
3) Measurements
4) Probabilities
5) Composition
Theories with modified measurements
Alternative probabilistic structure
Outcome Probability Function
Alternative probabilistic structure
Measurement
1) States
2) Transformations
3) Measurements
4) Probabilities
5) Composition
Theories with modified measurements
Theories with modified measurements
● A measurement postulate is a set of OPFs ● for every ● Any subset which sum to 1 form a measurement● In the case of quantum theory we have
1) States
2) Transformations
3) Measurements
4) Probabilities
5) Composition
Theories with modified measurements
1) States
2) Transformations
3) Measurements
4) Probabilities
5) Composition
Theories with modified measurements
Probabilistic structure
1) States
2) Transformations
3) Measurements
4) Probabilities
5) Composition
Theories with modified measurements
Probabilistic structureDynamical
Structure
1) States
2) Transformations
3)
4)
5) Composition
Quantum theory
Warning!
Changing the measurement postulates will change the structure of mixed states
What are mixed states?
Mixed states
Mixed states are equivalence classes of indistinguishable ensembles.
Indistinguishability is relative to the available measurements.
Changing the measurements will change which ensembles are indistinguishable.
State spaces
Systems with alternative measurement postulates have same pure states but different mixed states.
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Mixed states: rebit
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Mixed states
● If we allowed different measurements, these two ensembles may be distinguishable.
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Alternative rebit
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Alternative rebit
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Alternative rebit
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Alternative rebit
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Alternative rebit
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Mixed states
● We can no longer represent mixed states by density matrices when we change the measurements.
● Need to re-derive the mixed states
1) States
2) Transformations
3) Measurements
4) Probabilities
5) Composition
Theories with modified measurements
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Deriving the state space
● Find a set of fiducial measurements in
● Quantum theory
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Deriving the state space
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Mixed states
● The pure states of quantum theory correspond to rays in
● Every embedding of these pure states into gives a system with same pure states but different measurements
● So pure state structure of quantum theory does not fix the measurements.
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Classifying all alternatives to the Born rule
● So can classify all alternatives to the measurements postulates
● All state spaces with same pure states but different mixed states as quantum theory
● All possible embeddings of (as a manifold) in
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Representation theory and the Bloch sphere
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Probabilistic representation of transformations
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Probabilistic representation of transformations
Representation of SU(d)
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● Alternative measurement postulate implies different mixed states
● Different representations act on these different mixed states
● Alternative measurement postulate in correspondence with representation of SU(d)
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1) States
2) Transformations
3) Measurements
4) Probabilities
5) Composition
Theories with modified measurements
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Composite systems
● Systems A and B and AB.●
●
● Measurement on system A and B should correspond to a joint measurement on AB.
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Joint measurements
A
B
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Joint measurements
AB
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What about composite systems?
● Need a product between systems, and specifically between the OPFs:
● This product needs to obey certain constraints
● Other constraints also follow from operational considerations.
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The quantum case:
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Associativity
A
B
C
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Associativity
A
B
C
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Associativity
A
B
C
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Associativity
● Subjective groupings of devices lead to the same operational predictions
● “pre-operational” constraint
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What about composite systems?
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The result
● The only OPFs which obey this associativity constraint are the quantum ones.
● Under the assumption that (as a vector space) these sets are finite dimensional.
● Proof makes use of the linear action of the unitary group on
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The theorem
The only measurement postulate satisfying the “possibility of state estimation” has OPFs and
-product of the form
for all and where the operator satisfies and analogously for G.
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Conclusion
● The only GPT with the same dynamical structure as quantum theory is quantum theory itself
● The measurement postulates are redundant (assuming the operational framework)
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Other approaches
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Other approaches
225
Conclusion
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Perspectives on quantum theory
● Quantum logic● General probabilistic theory● Many others: Qubism, process theories...● Also many non-operational approaches
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How to study quantum theory
● We have seen how to study GPTs and non-classical logics.
● Quantum theory is just one instance of a GPT or NCL
● What about other (non operational) approaches?● What is a general branching theory for instance?
228
Perspectives on quantum theory
● Different approaches provide different insights● In the case of studying postulates I have used two
different approaches: Quantum Logic and GPTs● We were able to compress the postulates in two
directions.● Is there a more succinct formulation of quantum
theory?
229
Perspectives on quantum theory
● There are multiple mathematical structures in quantum theory
● A lattice structure (of subspaces)● A convex structure (of state and effect spaces)● Other structures to be explored: categorical...
230
Perspectives on quantum theory
● There is a unique logic with the same measurement structure as quantum theory: quantum theory
● There is a unique GPT with the same dynamical structure as quantum theory: quantum theory
231
Perspectives on quantum theory
● There are redundancies within the postulates of quantum theory.
● But need a background framework within which we can define things and show these redundancies.
